Maslov idempotent probability calculus, I

Maslov's idempotent measures allow an optimization theory to be derived at the same level of generality as probability and stochastic process theory. One purpose of this work is to present the basic concepts of this $(\max,+)$-version of probability theory. Using this framework we will see that the Bellman optimality principle is the idempotent version of the classical Markov causality principle. Applications to optimal control problems, Hamilton–Jacobi equations, and mathematical morphology are discussed in the second part of this study.